Nested Hilbert schemes, local Donaldson-Thomas theory, and Vafa-Witten / Seiberg-Witten correspondence

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Artan Sheshmani
Harvard University, Center for Mathematical Sciences and Applications
Wed, 11/08/2017 - 2:00pm - 2:50pm
RH 440R

We report on the recent rigorous and general construction of the deformation-obstruction theories and virtual fundamental classes of nested (flag) Hilbert scheme of one dimensional subschemes of a smooth projective algebraic surface. The nested Hilbert scheme is a moduli space, which parametrizes a nested chain of configurations of curves and points in the ambient surface. This construction will provide one with a general framework to compute a large class of already known invariants, such as Poincare invariants of Okonek et al, or the reduced local invariants of Kool and Thomas in the context of their local surface theory. We show how to compute the generating series of deformation invariants associated to the nested Hilbert schemes, and via exploiting the properties of vertex operators, prove that in some cases they are given by modular forms. We finally establish a connection between the Vafa-Witten invariants of local-surface threefolds (recently analyzed also by Tanaka and Thomas) and such nested Hilbert schemes. This construction (via applying Mochizuki’s wall- crossing techniques) enables one to obtain a relation between the generating series of Seiberg-Witten invariants of the surface, the Vafa-Witten invariants, and some modular forms. This is joint work with Amin Gholampour and Shing-Tung Yau appeared in arXiv:1701.08902 and arXiv:1701.08899.